 Solution

Write $y_i=|x_i|$. If $n$ is odd, the sample median $t=T_n$ equals one observation, with equally many $y_i$ below and above it; the corresponding terms $\mathrm{sign}(y_i/t-1)$ cancel and the median term is zero. If $n$ is even, the conventional median lies strictly between the two middle observations when the $y_i$ are distinct, so exactly half the terms are $-1$ and half are $+1$. In either caseHence $T_n$ is a Scale M-estimator.